Celebratio Mathematica

J. S. Birman and R. F. Williams: “Knotted periodic orbits in dynamical systems, I: Lorenz’s equations,” Topology 22 : 1 (1983), pp. 47–82. Part II was published in Low-dimensional topology (1983). MR 682059 Zbl 0507.58038 article

J. S. Birman and R. F. Williams: “Knotted periodic orbits in dynamical system, II: Knot holders for fibered knots,” pp. 1–60 in Low-dimensional topology. Edited by S. J. Lomonaco $J r .$ . Contemporary Mathematics 20. American Mathematical Society (Providence, RI), 1983. Part I was published in Topology 22:1 (1983). MR 718132 Zbl 0526.58043 incollection

Joan S. Birman	Related	Volume
Samuel James Lomonaco, Jr.	Related

R. F. Williams: “The braid index of an algebraic link,” pp. 697–703 in Braids (Santa Cruz, CA, 13–26 July 1986). Edited by J. S. Birman and A. Libgober. Contemporary Mathematics 78. American Mathematical Society (Providence, RI), 1988. MR 975103 Zbl 0673.57003 incollection

Until recently the braid index of an oriented link ($ = $ the minimal number of strands the link can be presented as a closed braid on) was essentially non-computable and hence a useless concept. But with the advent of the Jones polynomial [1985] and its relatives [Freyd et al. 1985; Lickorish and Millett 1985] computations are now possible. Jones himself has computed [1987] this invariant for all knots with ten or fewer crossings. In particular, a long outstanding conjecture on positive braids with a full twist (“very positive braids”) bas now been proved independently by Morton [1985] and Franks-Williams [Franks and Williams 1985]. Here we show that this theorem suffices to compute the braid index of an important family of links — the algebraic links. Briefly, at a singular point $ (w_0,z_0) $ of the variety $ P(w,z) = 0 $, $ w, z \in \mathbf{C} $ the complex numbers, the boundary $ \mathbb{S}^3 $ of a small neighborhood of $ (w_0,z_0) $ intersects the variety in an oriented one-manifold, $ L $. Then $ L \subset \mathbb{S}^3 $ is an oriented link. Any such link $ L $ is called an algebraic link. They have been studied for some time and are known to be positive braids. (i.e., can be presented as a closed braid with all crossings of the same sign — say positive or right-handed).

Anatoly S. Libgober	Related
Joan S. Birman	Related	Volume

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